Katok's polynomial growth conjecture for generalized diagonals in polygonal billiards

Let $M$ be a polygon, let $P_n$ denote the number of generalized diagonals of $M$ whose discrete length is at most $n$, and let $\epsilon>0$. Katok's conjecture. For every polygon $M$ and every $\epsilon>0$, there is a constant $C(M,\epsilon)$ such that

$$P_n<C(M,\epsilon)n^{2+\epsilon}.$$

This conjecture seeks an explicit sub-exponential bound for the growth of generalized diagonals in arbitrary, including irrational-angled, polygonal billiards. Masur's quadratic estimates settle the corresponding order of growth for rational-angled polygons, but the general case is described as a difficult open problem.